Todd W Neller Dept of Computer Science Dudo According to legend King Atahualpa of the Incas taught this to the Spanish conquistador Pizarro more than 400 years ago Mohr 1997 Origin generally believed to be 15 ID: 226497 Download Presentation

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Todd W. Neller. Dept. of Computer Science. Dudo. “According to legend, King Atahualpa of the Incas taught this to the Spanish conquistador Pizarro more than 400 years ago…” (Mohr, 1997). Origin generally believed to be 15.

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Slide1

Dudo: An Introduction to the Incan Bluffing Dice Game

Todd W. Neller

Dept. of Computer ScienceSlide2

Dudo

“According to legend, King Atahualpa of the Incas taught this to the Spanish conquistador Pizarro more than 400 years ago…” (Mohr, 1997)

Origin generally believed to be 15

th

c. Inca

Variants long popular in many Latin American countriesSlide3

Beyond Latin America

Many variants/names (Liar’s Dice, Perudo, Bluff, Call My Bluff, Cacho, Cachito)

Internationally popular

BoardGameGeek.com rank 236/49160 (top ½%!), 11/2/10Slide4

Dudo Overview

Bluffing dice game for 2+ players

Each player rolls 5 dice concealed under cup

Players make successively bolder claims about

all

dice rolled until player challenges

Loser of challenge loses dice

Last player with dice winsSlide5

Dudo ClaimsSlide6

Dudo Rules

Players each roll and privately view 5 dice concealed under a cup. 1’s are wild.

Players make successively greater claims until one challenges the previous claim with “Dudo!” (Sp. “I doubt it!”), all reveal dice, and:

More/less than claimed? Challenger/claimant loses dice according to difference.

Claim exactly correct? Challenger loses 1 die.

The next round begins with challenge winner.

The last player with dice wins.Slide7

Computer Play

Gather

around lab computers. (Carefully fold down desks.)

Commands:

cp ~

tneller

/pub/java/

dudo

/*.java

.

j

avac

*.java

java

DudoGame

Input claim format: “

dudo

” or “<num> <rank>” (e.g. “7 6”)

What do you observe?Slide8

Simple Dudo Player

SimpleDudoPlayer

follows a simple policy based on roll probabilities.

Make the strongest legal claim that is correct with probability ≥ ½.

If no such claim exists, call “

dudo

”. Slide9

Calculating Claim Probabilities

Example: What is the probability of exactly 2

of 5 dice being 6’s?

1’s are wild, so the probability of a single die being a 6 is 1/3.

6

5

different 5 dice rolls (in sequence)

5 choose 2 = 5! / (2! 3!) = 120 / (2 * 6) = 10 different ways of choosing 2 from 5

Each occurrence has prob. (1/3)

2

(1-1/3)

3

Answer:

10 ×

(1/3)

2

(1-1/3)

3

= 80/243Slide10

Calculating Claim Probabilities (cont.)

Example: What is the probability of

at least

2

of 5 dice being 6’s?

Exactly

0: 1

×

(1/3)

0

(1-1/3)

5

= 32/243

Exactly

1: 5

×

(1/3)

1

(1-1/3)

4

= 80/243

Exactly

2:

10 ×

(1/3)

2

(1-1/3)

3

= 80/243

Exactly 3: 10

×

(1/3)

3

(1-1/3)

2

= 40/243

Exactly

4: 5

×

(1/3)

4

(1-1/3)

1

= 10/243

Exactly 5:

1

×

(1/3)

5

(1-1/3)

0

= 1/243

At

least 2: (

80 + 40 + 10 +

1) / 243 = 131/243 ≈ .539 Slide11

Choosing a Claim

Example computation:

SimpleDudoPlayer

chooses the strongest claim

that is correct with probability ≥ ½:

five 6’sSlide12

Game Information

Dudo is a game of information:

State information: What you hold, how many dice others have

Probabilistic information: Roll probabilities

“Tell” information: Behavior indicators

Play information: Other player actions

Now that you know how

SimpleDudoPlayer

makes decisions, exploit that information.Slide13

Mixed Strategy

Predictability

Exploitability

Are these also examples of games with optimal mixed strategy?

Tic

Tac

Toe

Rock Scissors Paper

Blackjack / Pig

Poker

Clue

Game-theoretic techniques help us

compute or

approximate optimal mixed strategiesSlide14

Conclusion

Dudo is an excellent, ancient bluffing dice game.

Simple

application of probability and combinatorial mathematics yields a reasonable simple player

.

Predictability

Exploitability

Because player choices yield information, games of information often call for mixed strategies.Slide15

References

Reiner

Knizia

.

Dice Games Properly Explained

. Elliot Right-Way Books, Brighton Road, Lower

Kingswood

,

Tadworth

, Surrey, KT20 6TD UK, 1999

Merilyn

Simonds

Mohr.

The New Games Treasury

– More Than 500 Indoor and Outdoor Favorites with Strategies, Rules and Traditions,

Houghton Mifflin Co., Boston, 1997

Dudo rules online:

http://cs.gettysburg.edu/~tneller/games/rules/dudo.pdf

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